考虑损伤效应的正交各向异性板的弹塑性后屈曲分析

基于弹塑性力学和损伤理论,建立了一个与应力球张量有关的正交各向异性材料的混合硬化屈服准则,该准则无量纲化后与各向同性材料的Mises准则同构,进而建立了混合硬化正交各向异性材料的增量型弹塑性损伤本构方程和损伤演化方程．基于经典非线性板理论,得到了考虑损伤效应的正交各向异性板的增量型非线性平衡方程,且采用有限差分法和迭代法进行求解．数值算例中,讨论了损伤演化、初始缺陷对正交各向异性板弹塑性后屈曲行为的影响．数值结果显示了弹塑性后屈曲与弹性后屈曲的不同,并且损伤和损伤演化对板的弹塑性后屈曲的影响不可忽略．     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

Dynamic stability of cylindrical orthotropic shells with an elastic core under a longitudinal periodic load

The dynamic instability of a cylindrical orthotropic shell with an elastic core subjected to a longitudinal periodic load is considered. Equations are obtained for determining the regions of dynamic instability for different core models.     

Vibration analysis of a thermo-mechanically coupled large-scale welded wall based on an equivalent model

A vibration analysis method of a thermo-mechanically coupled large-scale welded wall is developed considering large-displacement. Firstly, the welded wall is theoretically normalized to an orthotropic thin plate, where the equivalent geometric and material parameters are derived in the light of the stress-function approach and the deformation compatibility conditions. Secondly, the equivalent heat conduction parameters are derived according to the heat transfer equation. Based on the partial differential equation of the heat conduction and the dynamic equilibrium equation of the thin plate, a thermo-mechanically coupled dynamic model of the equivalent orthotropic thin plate is established. Finally, numerical calculations are performed to discuss the influence of the various parameters on the thermo-mechanical responses adopting the Galerkin's method and the Runge–Kutta technique.     

考虑高阶横向剪切正交各向异性板非线性弯曲的微分求积分析

采用微分求积方法（DQ方法）讨论了计及高阶横向剪切的正交各向异性弹性板的非线性弯曲问题．导出了非线性控制方程的DQ形式,利用推广的DQWB技巧处理了高阶矩的边界条件．进一步推广并运用新的分析技术简化了非线性方程的计算．为说明该方法的可靠性和有效性,将考虑剪切变形及不计剪切变形的薄板的数值结果与三维弹性解析解及其它数值解进行了比较,同时研究了数值结果的收敛性,并考察了不同的节点分布对收敛速度的影响A·D2还考察了几何、材料参数及横向剪切效应对正交各向异性板非线性弯曲的影响．分析结果表明横向剪切效应对正交各向异性中厚板的影响是显著的．     

任意厚度具有自由边叠层板的精确解析解

自由边问题一直是三维弹性力学中的难题,通常很难满足自由边上一个正应力和两个剪应力都等于0．基于三维弹性力学基本方程和状态空间方法,引入自由边界位移函数并考虑全部弹性常数,建立了正交异性具有自由边单层和叠层板的状态方程．对状态方程中的变量以级数形式展开,通过边界条件的满足精确求解任意厚度具有自由边叠层板的位移和应力,此解满足层间应力和位移的连续条件．算例计算表明,采用引入的位移函数形式,简化了计算过程并且采用较少的级数项可以获得收敛解．与有限元方法计算结果进行了对比,可以得到较高精度的数值结果．其解可以作为其它数值方法和半解析方法的参考解．     

正交异性矩形板后屈曲摄动分析

本文从正交异性板Kármán型大挠度方程出发,以挠度为摄动参数,采用直接摄动法,研究了正交异性矩形板在面内压缩作用下的后屈曲性态.本文讨论了两种面内边界条件,同时考虑了初始挠度的影响.本文给出了多种复合材料板的计算结果.所得结果与实验结果的比较表明二者是一致的.     

Reflection and transmission of elastic waves in the multilayered orthotropic couple-stressed plates sandwiched between two elastic half-spaces

In this paper, a high efficient computational approach, the extended Legendre orthogonal polynomial method (LOPM) is provided to investigate the reflection and transmission of elastic waves in orthotropic couple-stress layered plates sandwiched between two elastic half-spaces. In this approach, the anisotropic couple-stress theory is introduced into the LOPM to calculate the reflection and transmission coefficients of the orthotropic interlayers. The stress components, couple-stress components, rotation vectors and governing equations are derived in terms of the Legendre orthogonal polynomial. The present method does not require calculations of the displacement solutions of each partial wave in anisotropic multilayered microstructures, but expands the displacement vector of each layer into a Legendre orthogonal polynomial series with the expansion coefficients to be calculated. The incident P wave and SH wave are calculated, respectively. The effects of length scale parameters in three different directions are studied. It is found that the reflection and transmission coefficients of the incident P wave are only related to the length scale parameter in the z direction. For incident SH wave, the influence of the length scale parameter along the thickness direction is much more significant than that of the length scale parameter in the y direction.     

The effect of transverse shear upon the buckling stability of inelastic orthotropic plates

The problem of buckling of inelastic orthotropic plates is analyzed. The phenomena associated with the transverse strains of the plate are considered in the framework of the Ambartsumyan theory of plates, which excludes the hypothesis of undeformed normals. The problem is solved with the aid of the localized strains theory, and it is shown that in the case of orthotropic materials, i.e., those characterized by a large E/G ratio, the effect of the transverse shear must be taken into account in determining the critical stress for a plate under compressive loads parallel to its surface.Mekhanika Polimerov, Vol. 1, No. 2, pp. 133–136, 1965     

Dynamic stability of an elastic orthotropic cylindrical shell with allowance for transverse shears

The resolvents for the dynamic stability of an elastic orthotropic cylindrical shell are obtained in accordance with the Ambartsumyan and Timoshenko-type refined theories. The regions of instability given by the classical and refined theories are compared. The dependence of the refinements on the shell parameters, the shear moduli of the material, and the buckling modes are investigated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 312–320, March–April, 1973.     

Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method

The subject of this article is solving free vibration problems of isotropic and orthotropic rectangular plates with linearly varying thickness along one direction. For the numerical solution to evaluate the frequencies of plates, the method of discrete singular convolution (DSC) is adopted. Frequency parameters are obtained for different types of boundary conditions, taper and aspect ratios. The effect of the mode number is also analyzed. The results obtained by the present numerical method show an excellent agreement with available published results.     

Behavior of cylindrical panels made of composite materials subjected to longitudinal impact in a gas flow

The dynamic properties of plates and cylindrical panels made of composite materials subjected to axial impact in a supersonic gas flow are investigated on the basis of a geometrically nonlinear orthotropic model and a linear wave equation for a one-dimensional medium. A solution is obtained by applying the Bubnov—Galerkin procedure with respect to the arc coordinate and a finite-difference method with respect to the axial coordinate and time. The aerodynamic pressure is found by means of an improved variant of linearized piston aerodynamics with a correction for curvature. Numerical results are presented and analyzed.     

A class of contact problems for thin anisotropic reinforced-plastic plates

Certain problems of the interaction of perfectly rigid smooth bodies and elastic reinforced-plastic plates are considered. The state of stress of the plate is described by the equations of orthotropic plates resisting interlaminar shear. Solutions are obtained in the case of a stamp in the shape of a rigid paraboloid of revolution indenting a circular plate with ring reinforcement and in the case of a parabolic cylinder indenting a unidirectionally reinforced plate in cylindrical bending. The expressions obtained for the contact pressure and the relation between the load P, the size of the contact region, and the settlement of the stamp are investigated with respect to their dependence on the orthotropy, wall thickness, and shear compliance parameters.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Ivano-Frankovskii Institute of Petroleum and Gas. Translated from Mekhanika Polimerov, No. 2, pp. 346–350, March–April, 1972.     

A Technique for Determining the Elastic and Dissipative Properties of Orthotropic Composites in Shear

A new method is presented for the characterization of three principal complex shear moduli of linear viscoelastic orthotropic materials, which is based on the measurement of complex torsional vibration frequencies of three rods of rectangular cross section. The rod-type test specimens are cut out from a composite plate along the principal material axes in the reinforcement plane. It is shown that the torsional stiffness of an elastic rod can be calculated not only by means of the Saint-Venant torsion theory, but also using a relationship obtained from the Reissner-Mindlin theory of plates. The transfer to a viscoelastic model of the material with complex moduli is realized with the help of the correspondence principle. By applying a numerical sensitivity analysis of natural frequencies to the shear moduli, the advisable width-to-thickness ratios of the specimens are found. As an illustration of data processing, the dynamic shear moduli and the loss factors for a GFRP fabric with an epoxy matrix are calculated. A comparison of the method offered with some known static and dynamic methods for determining the shear moduli of orthotropic materials is given.     

New exact series solutions for transverse vibration of rotationally-restrained orthotropic plates

The exact series solutions of plates with general boundary conditions have been derived by using various methods such as Fourier series expansion, improved Fourier series method, improved superposition method and finite integral transform method. Although the procedures of the methods are different, they are all Fourier-series based analytical methods. In present study, the foregoing analytical methods are reviewed first. Then, an exact series solution of vibration of orthotropic thin plate with rotationally restrained edges is obtained by applying the method of finite integral transform. Although the method of finite integral transform has been applied for vibration analysis of orthotropic plates, the existing formulation requires of solving a highly non-linear equation and the accuracy of the corresponding numerical results can be questionable. For that reason, an alternative formulation was proposed to resolve the issue. The accuracy and convergence of the proposed method were studied by comparing the results with other exact solutions as well as approximate solutions. Discussions were made for the application of the method of finite integral transform for vibration analysis of orthotropic thin plates.     

Refined Orthotropic Plate Motion Equations for Acoustoelasticity Problem Statement

The formulation of the acoustoelasticity problem is given on the basis of refined motion equations of orthotropic plates. These equations are constructed in the first approximation by reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of the theory of plates, where the approximation of the transverse tangential stresses and the transverse reduction stress is made with the help of trigonometric basis functions in the thickness direction. Wherein at the points of the boundary (front) surfaces, the static boundary conditions of the problem for tangential stresses are satisfied exactly and for transverse normal stress — approximately. Accounting for internal energy dissipation in the plate material is based on the Thompson—Kelvin—Voigt hysteresis model. In case of formulating problems on dynamic processes of plate deformation in vacuum, the equations are divided into two separate systems of equations. The first of these systems describes non-classical shear-free, longitudinal-transverse forms of movement, accompanied by a distortion of the flat form of cross sections, and the second system describes transverse bending-shear forms of movement. The latter are practically equivalent in quality and content to the analogous equations of the well-known variants of refined theories, but, unlike them, with a decrease in the relative thickness parameter, they lead to solutions according to the classical theory of plates. The motion of the surrounding the plate acoustic media is described by the generalized Helmholtz wave equations, constructed with account of energy dissipation by introducing into consideration the complex sound velocity according to Skudrzyk.     

Parametric vibrations of an orthotropic cylindrical shell with an elastic core. 2. Some numerical results

It is shown that the operator of the problem belongs to the class of oscillation operators. The first five regions of dynamic instability of an elastic orthotropic cylindrical shell with an elastic isotropic core are determined in the first approximation. The effect of the core and transverse shear in the shell on the width and location of these regions of dynamic instability of the system is determined. The effect of transverse shear on the natural frequencies of the empty shell is established.     

Singular integral equation method for 2D fracture analysis of orthotropic solids containing doubly periodic strip-like cracks on rectangular lattice arrays under longitudinal shear loading

The doubly periodic arrays of cracks represent an important mesoscopic model for analysis of the damage and fracture mechanics behaviors of materials. Here, in the framework of a continuously distributed dislocation model and singular integral equation approach, a highly accurate solution is proposed to describe the fracture behavior of orthotropic solids weakened by doubly periodic strip-like cracks on rectangular lattice arrays under a far-field longitudinal shear load. By fully comparing the current numerical results with known analytical and boundary element solutions, the high precision of the proposed solution is verified. Furthermore, the effects of periodic parameters and orthotropic parameter ratio on the stress intensity factor, crack tearing displacement, and effective shear modulus are studied, and an analytically polynomial estimation for the equivalent shear modulus is proposed in a certain range. The interaction distances among the vertical and horizontal periodic cracks are quite different, and their effects vary with the orthotropic parameter ratio. In addition, the dynamic problem is discussed briefly in the case where the material is subjected to harmonic longitudinal shear stress waves. Further work will continue the in-depth study of the dynamics problem of the doubly periodic arrays of cracks.     

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two-dimensional orthotropic thick plate theories with stress or mixed edge data.